Abstract

Estimation of particle-size distribution is analyzed for the complicated case of compound aerosols, in which particles are distinguished by sizes and optical constants. This task arises in a number of interesting practical situations when aerosol scatterers cannot be described with a common refractive index. This is an inverse problem with a large number of variables, and questions of formal inversion are of great importance here. They are discussed in detail, and an improved numerical-inversion method is proposed. The method provides a nonnegative and highly stable solution and makes it possible to include varied additional or a priori information. It is shown that the proposed technique is closely related to well-known linear and relaxation methods widely used in atmospheric optics. The algorithm for determination of bicomponent aerosol-size distribution is devised. It uses the intensity of light scattered at different angles and spectral-extinction measurements. In addition, the algorithm can incorporate a priori restrictions of size-spectra smoothness. A set of numerical examples illustrates the algorithm.

© 1995 Optical Society of America

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References

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  1. A. Tarantola, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation (Elsevier, Amsterdam, 1987), Chap. 1, p. 37.
  2. C. D. Rogers, “Statistical principles of inversion theory,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 117–134.
  3. S. L. Oshchepkov, O. V. Dubovik, “The informational content of a priori estimations in solving inverse problems of light scattering,” Opt. Atmos. 4, 88–95 (1991).
  4. W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.
  5. S. L. Oshchepkov, O. V. Dubovik, “Specific features of laser diffraction spectrometry method in conditions of anomalous diffraction,” J. Phys. D 26, 728–732 (1993).
    [CrossRef]
  6. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
    [CrossRef]
  7. B. L. Phillips, “A technique for numerical solution of certain integral equation of first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  8. V. F. Turchin, V. P. Kozlov, M. S. Malkevich, “The use of methods of mathematical statistics for the solution of the incorrect problems,” Usp. Fiz. Nauk 102, 345–386 (1970).
  9. M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
    [CrossRef]
  10. S. Twomey, “Comparison of constrained linear inverse and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
    [CrossRef]
  11. E. Trakhovsky, E. P. Shettle, “Improved inversion procedure for the retrieval of aerosol size distributions using aureole measurements,” J. Opt. Soc. Am. A 2, 2054–2061 (1985).
    [CrossRef]
  12. S. L. Oshchepkov, O. V. Dubovik, “The optimized iterative method for numerical solution of Fredholm integral equation of the first kind for positively defined values,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 165–172 (1994).
  13. J. M. Ortega, Introduction to Parallel and Vector Solution of Linear Systems (Plenum, New York, 1988), Chap. 2, p. 59.
  14. H. E. Fleming, “Comparison of linear inversion methods by examination of the duality between iterative and inverse matrix methods,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 325–355.
  15. O. V. Dubovik, S. L. Oshchepkov, T. V. Lapyonok, “An iteration-regularization method of non-linear inverse problems solution and its application for the interpretation of brightness spectra of water layer,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 106–113 (1994).
  16. J. M. Ortega, W. C. Reinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), Chap. 14, p. 504.
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.
  18. J. Lenoble, C. Brogniez, “A comparative review of radiation aerosol models,” Beitr. Phys. Atmos. 57, 1–20 (1984).
  19. O. Crepel, J-F. Gayet, J-F. Forttunol, “Project nephelometre: optimisation des caracteristiques optiques et de la methode a’echontillonnage,” Note 119 (Observatoire de Physique du Globe de Clemont-Ferrand, Laboratoire de Meteorologie Physique, Clemont-Ferrand, France, 1993).
  20. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5, p. 130.

1994 (2)

S. L. Oshchepkov, O. V. Dubovik, “The optimized iterative method for numerical solution of Fredholm integral equation of the first kind for positively defined values,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 165–172 (1994).

O. V. Dubovik, S. L. Oshchepkov, T. V. Lapyonok, “An iteration-regularization method of non-linear inverse problems solution and its application for the interpretation of brightness spectra of water layer,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 106–113 (1994).

1993 (1)

S. L. Oshchepkov, O. V. Dubovik, “Specific features of laser diffraction spectrometry method in conditions of anomalous diffraction,” J. Phys. D 26, 728–732 (1993).
[CrossRef]

1991 (1)

S. L. Oshchepkov, O. V. Dubovik, “The informational content of a priori estimations in solving inverse problems of light scattering,” Opt. Atmos. 4, 88–95 (1991).

1985 (1)

1984 (1)

J. Lenoble, C. Brogniez, “A comparative review of radiation aerosol models,” Beitr. Phys. Atmos. 57, 1–20 (1984).

1975 (1)

S. Twomey, “Comparison of constrained linear inverse and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

1970 (1)

V. F. Turchin, V. P. Kozlov, M. S. Malkevich, “The use of methods of mathematical statistics for the solution of the incorrect problems,” Usp. Fiz. Nauk 102, 345–386 (1970).

1968 (1)

1963 (1)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

1962 (1)

B. L. Phillips, “A technique for numerical solution of certain integral equation of first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5, p. 130.

Brogniez, C.

J. Lenoble, C. Brogniez, “A comparative review of radiation aerosol models,” Beitr. Phys. Atmos. 57, 1–20 (1984).

Chahine, M. T.

Crepel, O.

O. Crepel, J-F. Gayet, J-F. Forttunol, “Project nephelometre: optimisation des caracteristiques optiques et de la methode a’echontillonnage,” Note 119 (Observatoire de Physique du Globe de Clemont-Ferrand, Laboratoire de Meteorologie Physique, Clemont-Ferrand, France, 1993).

Dryard, D.

W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.

Dubovik, O. V.

O. V. Dubovik, S. L. Oshchepkov, T. V. Lapyonok, “An iteration-regularization method of non-linear inverse problems solution and its application for the interpretation of brightness spectra of water layer,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 106–113 (1994).

S. L. Oshchepkov, O. V. Dubovik, “The optimized iterative method for numerical solution of Fredholm integral equation of the first kind for positively defined values,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 165–172 (1994).

S. L. Oshchepkov, O. V. Dubovik, “Specific features of laser diffraction spectrometry method in conditions of anomalous diffraction,” J. Phys. D 26, 728–732 (1993).
[CrossRef]

S. L. Oshchepkov, O. V. Dubovik, “The informational content of a priori estimations in solving inverse problems of light scattering,” Opt. Atmos. 4, 88–95 (1991).

Edie, W. T.

W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.

Fleming, H. E.

H. E. Fleming, “Comparison of linear inversion methods by examination of the duality between iterative and inverse matrix methods,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 325–355.

Forttunol, J-F.

O. Crepel, J-F. Gayet, J-F. Forttunol, “Project nephelometre: optimisation des caracteristiques optiques et de la methode a’echontillonnage,” Note 119 (Observatoire de Physique du Globe de Clemont-Ferrand, Laboratoire de Meteorologie Physique, Clemont-Ferrand, France, 1993).

Gayet, J-F.

O. Crepel, J-F. Gayet, J-F. Forttunol, “Project nephelometre: optimisation des caracteristiques optiques et de la methode a’echontillonnage,” Note 119 (Observatoire de Physique du Globe de Clemont-Ferrand, Laboratoire de Meteorologie Physique, Clemont-Ferrand, France, 1993).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5, p. 130.

James, F. E.

W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.

Kozlov, V. P.

V. F. Turchin, V. P. Kozlov, M. S. Malkevich, “The use of methods of mathematical statistics for the solution of the incorrect problems,” Usp. Fiz. Nauk 102, 345–386 (1970).

Lapyonok, T. V.

O. V. Dubovik, S. L. Oshchepkov, T. V. Lapyonok, “An iteration-regularization method of non-linear inverse problems solution and its application for the interpretation of brightness spectra of water layer,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 106–113 (1994).

Lenoble, J.

J. Lenoble, C. Brogniez, “A comparative review of radiation aerosol models,” Beitr. Phys. Atmos. 57, 1–20 (1984).

Malkevich, M. S.

V. F. Turchin, V. P. Kozlov, M. S. Malkevich, “The use of methods of mathematical statistics for the solution of the incorrect problems,” Usp. Fiz. Nauk 102, 345–386 (1970).

Ortega, J. M.

J. M. Ortega, W. C. Reinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), Chap. 14, p. 504.

J. M. Ortega, Introduction to Parallel and Vector Solution of Linear Systems (Plenum, New York, 1988), Chap. 2, p. 59.

Oshchepkov, S. L.

O. V. Dubovik, S. L. Oshchepkov, T. V. Lapyonok, “An iteration-regularization method of non-linear inverse problems solution and its application for the interpretation of brightness spectra of water layer,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 106–113 (1994).

S. L. Oshchepkov, O. V. Dubovik, “The optimized iterative method for numerical solution of Fredholm integral equation of the first kind for positively defined values,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 165–172 (1994).

S. L. Oshchepkov, O. V. Dubovik, “Specific features of laser diffraction spectrometry method in conditions of anomalous diffraction,” J. Phys. D 26, 728–732 (1993).
[CrossRef]

S. L. Oshchepkov, O. V. Dubovik, “The informational content of a priori estimations in solving inverse problems of light scattering,” Opt. Atmos. 4, 88–95 (1991).

Phillips, B. L.

B. L. Phillips, “A technique for numerical solution of certain integral equation of first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Reinboldt, W. C.

J. M. Ortega, W. C. Reinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), Chap. 14, p. 504.

Rogers, C. D.

C. D. Rogers, “Statistical principles of inversion theory,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 117–134.

Roos, M.

W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.

Sadoulet, B.

W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.

Shettle, E. P.

Tarantola, A.

A. Tarantola, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation (Elsevier, Amsterdam, 1987), Chap. 1, p. 37.

Trakhovsky, E.

Turchin, V. F.

V. F. Turchin, V. P. Kozlov, M. S. Malkevich, “The use of methods of mathematical statistics for the solution of the incorrect problems,” Usp. Fiz. Nauk 102, 345–386 (1970).

Twomey, S.

S. Twomey, “Comparison of constrained linear inverse and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

Beitr. Phys. Atmos. (1)

J. Lenoble, C. Brogniez, “A comparative review of radiation aerosol models,” Beitr. Phys. Atmos. 57, 1–20 (1984).

Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana (2)

O. V. Dubovik, S. L. Oshchepkov, T. V. Lapyonok, “An iteration-regularization method of non-linear inverse problems solution and its application for the interpretation of brightness spectra of water layer,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 106–113 (1994).

S. L. Oshchepkov, O. V. Dubovik, “The optimized iterative method for numerical solution of Fredholm integral equation of the first kind for positively defined values,” Izv. Ross. Akad. Nauk Atmos. Fiz. Okeana 30, 165–172 (1994).

J. Assoc. Comput. Mach. (2)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

B. L. Phillips, “A technique for numerical solution of certain integral equation of first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, “Comparison of constrained linear inverse and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

S. L. Oshchepkov, O. V. Dubovik, “Specific features of laser diffraction spectrometry method in conditions of anomalous diffraction,” J. Phys. D 26, 728–732 (1993).
[CrossRef]

Opt. Atmos. (1)

S. L. Oshchepkov, O. V. Dubovik, “The informational content of a priori estimations in solving inverse problems of light scattering,” Opt. Atmos. 4, 88–95 (1991).

Usp. Fiz. Nauk (1)

V. F. Turchin, V. P. Kozlov, M. S. Malkevich, “The use of methods of mathematical statistics for the solution of the incorrect problems,” Usp. Fiz. Nauk 102, 345–386 (1970).

Other (9)

J. M. Ortega, Introduction to Parallel and Vector Solution of Linear Systems (Plenum, New York, 1988), Chap. 2, p. 59.

H. E. Fleming, “Comparison of linear inversion methods by examination of the duality between iterative and inverse matrix methods,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 325–355.

W. T. Edie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971), Chap. 8, p. 155.

A. Tarantola, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation (Elsevier, Amsterdam, 1987), Chap. 1, p. 37.

C. D. Rogers, “Statistical principles of inversion theory,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 117–134.

J. M. Ortega, W. C. Reinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), Chap. 14, p. 504.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.

O. Crepel, J-F. Gayet, J-F. Forttunol, “Project nephelometre: optimisation des caracteristiques optiques et de la methode a’echontillonnage,” Note 119 (Observatoire de Physique du Globe de Clemont-Ferrand, Laboratoire de Meteorologie Physique, Clemont-Ferrand, France, 1993).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5, p. 130.

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Figures (13)

Fig. 1
Fig. 1

Comparison of the size-distribution retrievals obtained by the use of different methods for a monocomponent aerosol, a monomodal distribution, size range 0.1 μm < r < 10 μm, and n = 20.

Fig. 2
Fig. 2

Comparison of the size-distribution retrievals obtained by the use of different methods for a monocomponent aerosol, a bimodal distribution, size range 0.002 μm < r < 25 μm, and n = 20.

Fig. 3
Fig. 3

Comparison of the size-distribution retrievals obtained by the use of different methods for a compound aerosol containing two components distinguished by refractive indices and size distributions.

Fig. 4
Fig. 4

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from angular light scattering (upper graphs) and angular light scattering and spectral extinction (lower graphs)

Fig. 5
Fig. 5

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from angular light scattering with smoothness restrictions, with γsm1 = γsm2 = 0.003 and γext = 0 (upper graphs), and angular light scattering and spectral extinction with smoothness restrictions, with γsm1 = γsm2 = 0.003 and γext = 1 (lower graphs).

Fig. 6
Fig. 6

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from angular light scattering and spectral extinction with smoothness restrictions. γsm1 = γsm2 = 0.003, γext = 1.

Fig. 7
Fig. 7

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from angular light scattering and spectral extinction with smoothness restrictions. γsm1 = γsm2 = 0.003, γext = 1.

Fig. 8
Fig. 8

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from angular light scattering and spectral extinction with smoothness restrictions. γsm1 = γsm2 = 0.003, γext = 1.

Fig. 9
Fig. 9

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from angular light scattering and spectral extinction with smoothness restrictions. γsm1 = γsm2 = 0.003, γext = 1.

Fig. 10
Fig. 10

Comparison of the size-distribution retrievals from a limited set of angular light-scattering data obtained by the use of different methods for a monocomponent aerosol (same as in Fig. 1).

Fig. 11
Fig. 11

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from limited angular light scattering data, with γ = 0 (upper graphs), and angular light scattering and spectral extinction, with γext = 1 and γsm = 0.

Fig. 12
Fig. 12

Comparison of real aerosol spectra and spectra estimated by the use of Eq. (30) from limited angular light-scattering data with smoothness restrictions, with γsm1 = γsm2 = 0.003 and γext = 0 (upper graphs), and angular light scattering and spectral extinction with smoothness restrictions, with γsm1 = γsm2 = 0.003 and γext = 1.

Fig. 13
Fig. 13

Comparison of real aerosol spectra and spectra obtained by the use of Eq. (30). The upper graphs present the same case as those in the lower part of Fig. 11, and the lower graphs present the same case as the lower part of Fig. 6. An absolute rather than a logarithmic scale is used here.

Equations (43)

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σ ( θ ) = l = 1 L - + σ ( k r , m ^ l , θ ) n l ( ln r ) d ln r ,
σ * = K φ + Δ ,
K T ( C σ * ) - 1 K φ = K T ( C σ * ) - 1 σ * ,
( φ ^ i - φ ^ i ) 2 = { [ K T ( C σ * ) - 1 K ] - 1 } i i = { Φ - 1 } i i ,
ρ i k = { Φ } i k ( { Φ } i i { Φ } k k ) 1 / 2 ,
( φ ^ i - φ ^ i ) 2 = { { Φ } i i [ 1 - ρ i , 12 ( i - 1 ) ( i + 1 ) n 2 ] } - 1 ,
1 - ρ i , 1 ( i - 1 ) ( i + 1 ) n 2 1 - ρ i , 1 ( i - 1 ) ( i + 1 ) ( n - n ) 2 = k = n - n + 1 n [ 1 - ρ i k , 1 ( i - 1 ) ( i + 1 ) ( k - 1 ) 2 ] ,
( K T K + γ I n ) φ = K T σ * + γ φ 0 ,
( K T K + γ Ω ) φ = K T σ * ,
[ K T K + ( C φ ) - 1 ] φ = K T σ * ,
σ * = K φ + Δ , β * = B φ + Δ 1 ,
[ K T ( C σ * ) - 1 K + B T ( C β * ) - 1 B ] φ = K T ( C σ * ) - 1 σ * + B T ( C β * ) - 1 β * ,
φ i p + 1 = φ i p ( σ i * σ i p ) ,
φ i p + 1 = φ i p j = 1 m [ 1 + ( σ j * σ j p - 1 ) K j i ] .
a p + 1 = a p - [ U p T ( C f * ) - 1 U p + D p T ( C d * ) - 1 D p ] - 1 × [ U p T ( C f * ) - 1 ( f p - f * ) + D p T ( C d * ) - 1 ( d p - d * ) ] ,
φ i p + 1 = φ i p - H ( K φ p - σ * ) ,
φ p + 1 = φ p - H [ Φ φ p - K T ( C σ * ) - 1 σ * - B T ( C β * ) - 1 β * ] .
{ H } i i = ( k = 1 n { Φ } i k ) - 1 ,
[ U p T ( C f * ) - 1 U p + D p T ( C d * ) - 1 D p ] Δ φ p = U p T ( C f * ) - 1 Δ f p + D p T ( C d * ) - 1 Δ d p ,
( Δ a p ) q + 1 = ( Δ a p ) q - H p [ Φ p ( Δ a p ) q - U p T ( C f * ) - 1 Δ f p + D p T ( C d * ) - 1 Δ d p ] ,
{ H p } i i = [ k = 1 n ( { Φ p } i k ) ] - 1 , Φ p = U p T ( C f * ) - 1 U p + D p T ( C d * ) - 1 D p .
a p + 1 = a p - t p ( U p T U p + γ p I n ) - 1 U p T ( f p - f * )
( Δ a p ) q q ( Δ a p ) ,
( Δ a p ) q ( Δ a p ) q + 1 ( Δ a p ) ,
a p + 1 = a p - t p H p [ U p T ( C f * ) - 1 ( f p - f * ) + D p T ( C d * ) - 1 ( d p - d * ) ] ,
σ sca * = ( K sca 1 K sca 2 ) ( φ 1 φ 2 ) + Δ sca , σ ext * = ( K ext 1 K ext 2 ) ( φ 1 φ 2 ) + Δ ext ,
{ K } j i l = ln r i l - 1 ln r i l ln r - ln r i l - 1 ln r i l - ln r i l - 1 σ ( ) 4 / 3 π r 3 d ln r + ln r i l ln r i l + 1 ln r i l + 1 - ln r ln r i l + 1 - ln r i l σ ( ) 4 / 3 π r 3 d ln r ,
( 0 1 0 2 ) = [ S 1 0 0 S 2 ] ( a 1 a 2 ) + Δ sml .
C f sca = I m sca 2 ,             C f ext = I m 1 ext 2 , C sm = [ sm 1 2 I ( n 1 - 2 ) 0 0 sm 2 2 I ( n 2 - 2 ) ] .
a p + 1 = a p - t p H p [ ( U sca p ) T ( f sca p - f sca * ) + γ ext ( U ext p ) T ( f ext p - f ext * ) + Ω a p ] ,
{ H p } i i = { k = 1 n 1 + n 2 [ { ( U sca p ) T ( U sca p ) + γ ext ( U ext p ) T ( U ext p ) + Ω i k } ] } - 1 δ i i ,
a p = ( a 1 p a 2 p ) ,             { a l } i 1 = ln ( { φ l } i 1 ) , U p = ( U 1 p U 2 p ) ,             Ω = [ γ sm 1 Ω n 1 0 0 γ sm 2 Ω n 2 ] .
{ U p } j i = f j a i | a p = ln σ j ln φ i | φ p = { K } j i { φ l p } i { σ p } j .
0 = [ ( f sca * - f sca ) T ( f sca * - f sca ) + γ ext ( f ext * - f ext ) T ( f ext * - f ext ) m + γ ext m 1 ] 1 / 2 ,
φ i p = φ i p + 1 { j = 1 m [ 1 + ( σ j * σ j p - 1 ) { U p } j i ] } t p ,
σ j p σ j * ,             [ ( σ j * / σ j p ) - 1 ] { U p } j i 0 ,
φ i p = φ i p + 1 ( exp { j = 1 m [ σ j * - σ j p σ j p { U p } j i ] } ) t p .
a p + 1 = a p - t q [ U p T ( f p - f * ) ] .
φ p + 1 = φ p - H ( Φ φ p - Φ φ ) = φ p - H ( Y p - Y * ) ,
Δ φ l + 1 = φ l + 1 - φ = φ l - φ - H ( Φ φ l - Φ φ ) = ( I n - H Φ ) Δ φ l , Δ Y l + 1 - Φ φ l + 1 - Y * = Φ φ l - Y * - Φ H ( Φ φ l - Y * ) = ( I n - Φ H ) Δ Y l .
Δ Y l + 1 = ( I n - Φ H ) l Δ Y 0 ,             Δ φ l + 1 = ( I n - H Φ ) l Δ φ 0 ,
T r i = δ r i - Φ i i 1 / 2 ρ r i k = 1 n ( Φ k k ρ k i ) ,
T l = T - I n - H Φ ,             l 1.

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